1 + x \le e^x, x \in R (1)

on an extension line of the construction of e. The distinction of our method is to prove the inequality (1) without using the differential calculus. As is shown, the inequality(1) characterizes the exponential funtion $e^x$. Furthermore, we can prove some fundamental properties of $e^ x$ by using (1) and find that (1) is extremely important inequality with respect to the exponential functin $e^x$.